Non-Classical Lagrangian Dynamics and Potential Maps
Keywords:
ultra-harmonicmap,ultra-potentialmap,Lagrangian,Hamiltonian,Lorentz-Udristeworldforcelaw.Abstract
The basic theory regarding Nonclassical Lagrangian Dynamicsand Potential Mapswasannouncedin [7].Sinceitsmathematicalimpactisnowatlargevogue,wereinforcesomearguments.Section1extendsthe theoryofharmonicandpotentialmapsinthelanguageofdifferentialgeometry.Section2definesageneralized Lorentzworld- forcelawan dshowsthatany PDEsystemoforderone(inparticular,p-flow)generat essuchalaw inasuitablegeometricalstructure.Inotherwords,thesolutionsofanyPDEsystemoforderoneareharmonicor
potentialmaps,i.e.,theyaresolutionsofEuler-LagrangeprolongationPDEsystemofordertwobuiltviaRiemann-LagrangestructuresandaleastsquaresLagrangian.Section3formulatesopenproblemsregardingthegeometryof semi-Riemannmanifolds(J1(T,M),S1),(J2(T,M),S2).Section4showsthattheLorentz-Udristeworld-force
lawisequivalenttocertaincovariantHamiltonPDEson(J1(T,M),S1).Section5describesthemapsdetermining
acontinuousgroupoftransformationsasultra-potentialmaps.
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